We follow the similar argument in [[1]}, Section 4.1] for our relative case, see also [[2]}, [3]}, [4]}].
Denote by \(\mathcal {Z}\subset \mathbb {P}\times _{S}\widetilde{\mathcal {Q}}\)  the closed subscheme consisting of all points \(([a], [q])\)  such that for any geometric point \(s\)  of \(S\) , \(q_{s}\circ (a_{s}\otimes \mathrm {id})\)  factors through \(\widetilde{\mathrm {ev}}_{s}\)  in a commutative diagram as follows
\({H^0(X_{s},F_{\mathcal {E}_{s}}(\mathcal {F}_{0,s})(m))\otimes \mathcal {E}_{s}\otimes \pi ^*\mathcal {O}_{X_{s}}(-m)[d]_{a_{s}\otimes \mathrm {id}} [r]^-{\widetilde{\mathrm {ev}}_{s}} & \mathcal {F}_{0,s}[d]^{\varphi _{s}}\\V\otimes \mathcal {E}_{s}\otimes \pi ^*\mathcal {O}_{X_{s}}(-m)[r]_-{q_{s}} & \mathcal {F}_s}\) 
